Superformula

The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003. Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis has filed a patent application related to the synthesis of patterns generated by the superformula, which expired effective 2020-05-10. In polar coordinates, with r the radius and \varphi the angle, the superformula is: r\left(\varphi\right) = \left( \left| \frac{\cos\left(\frac{m\varphi}{4}\right)}{a} \right| ^{n_2} + \left| \frac{\sin\left(\frac{m\varphi}{4}\right)}{b} \right| ^{n_3} \right) ^{-\frac{1}{n_{1}}}. By choosing different values for the parameters a, b, m, n_1, n_2, and n_3, different shapes can be generated. The formula was obtained by generalizing the superellipse, named and popularized by Piet Hein, a Danish mathematician.

The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003. Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis has filed a patent application related to the synthesis of patterns generated by the superformula, which expired effective 2020-05-10. In polar coordinates, with r the radius and \varphi the angle, the superformula is: r\left(\varphi\right) = \left( \left| \frac{\cos\left(\frac{m\varphi}{4}\right)}{a} \right| ^{n_2} + \left| \frac{\sin\left(\frac{m\varphi}{4}\right)}{b} \right| ^{n_3} \right) ^{-\frac{1}{n_{1}}}. By choosing different values for the parameters a, b, m, n_1, n_2, and n_3, different shapes can be generated. The formula was obtained by generalizing the superellipse, named and popularized by Piet Hein, a Danish mathematician. function sf2d(n, a) u = [0:.001:2 * pi]; raux = abs(1 / a(1) .* abs(cos(n(1) * u / 4))) .^ n(3) + abs(1 / a(2) .* abs(sin(n(1) * u / 4))) .^ n(4); r = abs(raux) .^ (- 1 / n(2)); x = r .* cos(u); y = r .* sin(u); plot(x, y); end x = r_1(\theta)\cos\theta \cdot r_2(\phi)\cos\phi, y = r_1(\theta)\sin\theta \cdot r_2(\phi)\cos\phi, z = r_2(\phi)\sin\phi, where \phi (latitude) varies between −π/2 and π/2 and θ (longitude) between −π and π. 3D superformula: a = b = 1; m, n 1 , n 2 and n 3 are shown in the pictures. Image:Sf3d_3257.svg Image:Sf3d_3.5.5.5.svg Image:Sf3d_3301515.svg Image:Sf3d_7284.svg Image:Sf3d_5111.svg Image:sf3d_4.5.54.svg Image:sf3d_8.5.58.svg Image:Sf3d_4121515.svg A GNU Octave program for generating these figures: function sf3d(n, a) u = [- pi:.05:pi]; v = [- pi / 2:.05:pi / 2]; nu = length(u); nv = length(v); for i = 1:nu for j = 1:nv raux1 = abs(1 / a(1) * abs(cos(n(1) .* u(i) / 4))) .^ n(3) + abs(1 / a(2) * abs(sin(n(1) * u(i) / 4))) .^ n(4); r1 = abs(raux1) .^ (- 1 / n(2)); raux2 = abs(1 / a(1) * abs(cos(n(1) * v(j) / 4))) .^ n(3) + abs(1 / a(2) * abs(sin(n(1) * v(j) / 4))) .^ n(4); r2 = abs(raux2) .^ (- 1 / n(2)); x(i, j) = r1 * cos(u(i)) * r2 * cos(v(j)); y(i, j) = r1 * sin(u(i)) * r2 * cos(v(j)); z(i, j) = r2 * sin(v(j)); endfor; endfor; mesh(x, y, z); endfunction; r\left(\varphi\right) = \left( \left| \frac{\cos\left(\frac{y\varphi}{4}\right)}{a} \right| ^{n_2} + \left| \frac{\sin\left(\frac{z\varphi}{4}\right)}{b} \right| ^{n_3} \right) ^{-\frac{1}{n_{1}}} This allows the creation of rotationally asymmetric and nested structures. In the following examples a, b, {n_2} and {n_3} are 1: Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization Least Squares Fitting of Chacón-Gielis Curves By the Particle Swarm Method of Optimization Superformula 2D Plotter & SVG Generator Interactive example using JSXGraph SuperShaper: An OpenSource, OpenCL accelerated, interactive 3D SuperShape generator with shader based visualisation (OpenGL3)

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